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A014571 Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.

%I A014571 #64 Feb 16 2025 08:32:33
%S A014571 4,1,2,4,5,4,0,3,3,6,4,0,1,0,7,5,9,7,7,8,3,3,6,1,3,6,8,2,5,8,4,5,5,2,
%T A014571 8,3,0,8,9,4,7,8,3,7,4,4,5,5,7,6,9,5,5,7,5,7,3,3,7,9,4,1,5,3,4,8,7,9,
%U A014571 3,5,9,2,3,6,5,7,8,2,5,8,8,9,6,3,8,0,4,5,4,0,4,8,6,2,1,2,1,3,3,3,9,6,2,5,6
%N A014571 Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.
%C A014571 This constant is transcendental (Mahler, 1929). - _Amiram Eldar_, Nov 14 2020
%D A014571 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.8 Prouhet-Thue-Morse Constant, p. 437.
%H A014571 Harry J. Smith, <a href="/A014571/b014571.txt">Table of n, a(n) for n = 0..20000</a>
%H A014571 Boris Adamczewski and Yann Bugeaud, <a href="https://www.jstor.org/stable/27642252">A short proof of the transcendence of Thue-Morse continued fractions</a>, The American Mathematical Monthly, Vol. 114, No. 6 (2007), pp. 536-540; <a href="http://irma.math.unistra.fr/~bugeaud/travaux/Monthly1.pdf">alternative link</a>.
%H A014571 Jean-Paul Allouche and Jeffrey Shallit, <a href="https://doi.org/10.1007/978-1-4471-0551-0_1">The ubiquitous Prouhet-Thue-Morse sequence</a>, in: C. Ding, T. Helleseth, and H. Niederreiter (eds.), Sequences and their applications, Springer, London, 1999, pp. 1-16; <a href="http://ciencias.uis.edu.co/lenguajes/doc/ubiq.pdf">alternative link</a>.
%H A014571 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p.726 ff.
%H A014571 Michel Dekking, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k5619975b/f21.image.r">Transcendance du nombre de Thue-Morse</a>, Comptes Rendus de l'Academie des Sciences de Paris, Série A, Vol. 285 (1977) A157-A160.
%H A014571 Arturas Dubickas, <a href="http://dx.doi.org/10.1016/j.jnt.2005.07.004">On the distance from a rational power to the nearest integer</a>, Journal of Number Theory, Volume 117, Issue 1, March 2006, Pages 222-239.
%H A014571 Kurt Mahler, <a href="https://doi.org/10.1007/BF01454845">Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen</a>, Mathematische Annalen, Vol. 101 (1929), pp. 342-366, <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002273217">alternative link</a>.
%H A014571 R. Schroeppel and R. W. Gosper, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/series.html#item122">HACKMEM #122</a> (1972).
%H A014571 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Thue-MorseConstant.html">Thue-Morse Constant</a>.
%H A014571 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A014571 Equals Sum_{k>=0} A010060(n)*2^(-(k+1)). [Corrected by _Jianing Song_, Oct 27 2018]
%F A014571 Equals Sum_{k>=1} 2^(-(A000069(k)+1)). - _Jianing Song_, Oct 27 2018
%F A014571 From _Amiram Eldar_, Nov 14 2020: (Start)
%F A014571 Equals 1/2 - (1/4) * A215016.
%F A014571 Equals 1/(3 - 1/A247950). (End)
%e A014571 0.412454033640107597783361368258455283089...
%e A014571 In hexadecimal, .6996966996696996... .
%p A014571 A014571 := proc()
%p A014571     local nlim,aold,a ;
%p A014571     nlim := ilog2(10^Digits) ;
%p A014571     aold := add( A010060(n)/2^n,n=0..nlim) ;
%p A014571     a := 0.0 ;
%p A014571     while abs(a-aold) > abs(a)/10^(Digits-3) do
%p A014571         aold := a;
%p A014571         nlim := nlim+200 ;
%p A014571         a := add( A010060(n)/2^n,n=0..nlim) ;
%p A014571     od:
%p A014571     evalf(%/2) ;
%p A014571 end:
%p A014571 A014571() ; # _R. J. Mathar_, Mar 03 2008
%t A014571 digits = 105; t[0] = 0; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1-t[(n-1)/2]; FromDigits[{t /@ Range[digits*Log[10]/Log[2] // Ceiling], -1}, 2] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 20 2014 *)
%t A014571 1/2-1/4*Product[1-2^(-2^k), {k, 0, Infinity}] // N[#, 105]& // RealDigits // First (* _Jean-François Alcover_, May 15 2014, after _Steven Finch_ *)
%t A014571 (* ThueMorse function needs $Version >= 10.2 *)
%t A014571 P = FromDigits[{ThueMorse /@ Range[0, 400], 0}, 2];
%t A014571 RealDigits[P, 10, 105][[1]] (* _Jean-François Alcover_, Jan 30 2020 *)
%o A014571 (PARI) default(realprecision, 20080); x=0.0; m=67000; for (n=1, m-1, x=x+x; x=x+sum(k=0, length(binary(n))-1, bittest(n, k))%2); x=10*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014571.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 25 2009
%o A014571 (PARI) 1/2-prodinf(n=0,1-1.>>2^n)/4 \\ _Charles R Greathouse IV_, Jul 31 2012
%Y A014571 Cf. A000069, A001969, A010060, A058631, A215016, A247950.
%K A014571 nonn,cons
%O A014571 0,1
%A A014571 _Eric W. Weisstein_
%E A014571 Corrected and extended by _R. J. Mathar_, Mar 03 2008